The Dutch scientist Christiaan Huygens
suggested a graphical method of predicting the future position of a
wavefront, knowing its current position. 

His principle is stated as follows. 

Each point on the existing wavefront can be considered
to act as a source of waves (sometimes referred to
as secondary wavelets). 


For physical evidence to support this suggestion, just consider
the phenomenon of diffraction of waves by a small aperture. 

When a small part of a plane wavefront is isolated, it does
behave like a point source... I rest my case (well, Huygens'
case)... 



In the following examples it is perhaps
easiest to imagine waves on the surface of water (as observed in a
ripple tank), but the results can be applied to any two (or three)
dimensional waves. 



First, a rather trivial example. 

Consider the set of plane (straight) waves shown below. 

The velocity of propagation of the waves is v. 

To apply the principle, we must pretend that we cannot guess
where the wavefront will be, say, t seconds later! 



First chose a point (any point), A, on the wavefront and draw
an arc of radius vt. 

This is the distance the secondary wavelets will have moved in
t seconds. 







Now chose another point, B, at random, on the wavefront and
repeat the process. 







The new wavefront is the tangent to the two curves…well what a
surprise, it's just where we expected! 

However, we will now consider two situations where the principle
can help make useful predictions. 



Reflection of Waves Using Huygens’ Principle 

Consider a set of plane waves moving towards a reflecting
surface, indicated by the line xx’. 

At time t = 0, point A on the wavefront reaches the reflecting
surface. 

The red arrow is a "ray" showing the direction of motion of the
waves. A ray is always at 90° to the wavefront. 







We will try to find the position of the wavefront at time t,
the instant when point B reaches the reflecting surface. 



First, draw an arc of radius equal to the distance B C (see next
diagram). 

The secondary wavelets from point A will have travelled this far
by the time the waves at point B reach point C. 

The new wavefront is the tangent to this arc which passes
through point C. 





Now, using the observed fact that the direction of propagation
of a wave is always at 90° to the wave front, we can predict the
direction of motion of the waves after reflection. 





The angle of incidence is the angle between the
direction of propagation of the waves and a normal to the reflecting
surface before reflection. 



The angle of reflection is the angle between the
direction of propagation of the waves and a normal to the reflecting
surface after reflection. 



We therefore wee that Huygens' method predicts that waves obey
the familiar law of reflection, easily observed using a
light beam and a mirror. 





Refraction of Waves Using Huygens’ Principle 

When waves travel across a boundary between two different media,
the speed of propagation changes. 

For example, the speed of light in a vacuum is
3×10^{8}ms^{1},
whereas in glass its speed is about 2×10^{8}ms^{1}. 

The change in speed can result in a change in direction of
propagation of the waves. 

This change in direction is called refraction. 



The diagrams below show how to use Huygens’ principle to predict
the position of the wavefront when waves move from a medium in
which they have speed v_{1} to a medium in which they have
speed v_{2}. 

In this case, v_{2} < v_{1} 

In these diagrams the line x x’ represents the boundary between
the two media. 







At time t = 0, point A on the wavefront reaches the boundary. 

Consider secondary wavelets emitted from A at time t = 0. 

At time t seconds later, point B reaches the boundary. 

At time t, the secondary wavelets emitted from A have moved a
distance v_{2}t. 



The position of the new wavefront is shown by line C D. 

The situation at a later time is shown in the next diagram. 







Notice that the change in speed of the waves inevitably produces
a change in the wavelength, as explained here. 
